Exercise 37 Page 627

The area of a rhombus is half the product of its diagonals. Recall that the diagonals of a rhombus are perpendicular. We also know that the diagonals bisect each other. That gives us that the two halves of the horizontal diagonal are congruent. By the definition of congruence, they have the same length.

Let's consider the right triangle with an acute angle of 45^(∘).

By using the tangent ratio we can calculate the length of the adjacent side of this triangle. Let's do it!

tan(θ)=Opp./Adj.

SubstituteValues

Substitute values

tan(45^(∘))=3/a

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Solve for a

MultEqn

LHS * a=RHS* a

tan(45^(∘))* (a)=3

DivEqn

.LHS /tan(45^(∘)).=.RHS /tan(45^(∘)).

a=3/tan(45^(∘))

UseCalc

Use a calculator

a=3

Now we know the length of the adjacent side. Once more we will use that the diagonals bisect each other.

By using the Segment Addition Postulate we can now find the lengths of the diagonals. d_1: 3+3= 6cm d_2: 3+3=6cm Using the diagonal lengths, let's find the area of the rhombus.

A=1/2d_1 d_2

SubstituteII

d_1= 6, d_2= 6

A=1/2( 6)(6)

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Evaluate right-hand side

MoveRightFacToNumOne

1/b* a = a/b

A=(6)(6)/2

CalcQuotProd

Calculate quotient and product

A=18

The area of the rhombus is 18cm^2.